By the Ostrogradsky–Hermite reduction [1]}, [2]}, [3]}, we can decompose a rational function \(f\in F(z)\) 
as \(f = D_z(g) + a/b\) , where \(g\in F(z)\)  and \(a, b \in F[z]\)  are such that \(\deg _z(a)<\deg _z(b), \gcd (a, b)=1\) ,
and \(b\)  is a squarefree polynomial in \(F[z]\) . By Lemma , \(f\)  is \(D_z\) -integrable in \(F(z)\)  if and only if \(a=0\) .
